On Calculating the Chromatic Symmetric Function

TL;DR

This paper introduces new methods for calculating the chromatic symmetric function of graphs in specific bases, providing combinatorial proofs and connecting existing algorithms to these new techniques.

Contribution

It presents a novel approach for computing the CSF in chromatic-bases from forests and an efficient method in the $m_\lambda$-basis, linking algebraic and combinatorial techniques.

Findings

New method for CSF in chromatic-bases from forests

Efficient calculation of CSF in $m_\lambda$-basis

Reinterpretation of existing algorithm as a special case

Abstract

This paper investigates methods for calculating the chromatic symmetric function (CSF) of a graph in chromatic-bases and the $m_\lambda$-basis. Our key contributions include a novel approach for calculating the CSF in chromatic-bases constructed from forests and an efficient method for determining the CSF in the $m_\lambda$-basis. As applications, we present combinatorial proofs for two known theorems that were originally established using algebraic techniques. Additionally, we demonstrate that an algorithm introduced by Aliste-Prieto, de Mier, Orellana, and Zamora can be viewed as a case of our proposed method.